Transducer dynamics

Citation
Transducer dynamics

Material Information

Title:
Transducer dynamics
Creator:
Dolzhenko, Egor
Place of Publication:
[Tampa, Fla]
Publisher:
University of South Florida
Publication Date:
Language:
English

Subjects

Subjects / Keywords:
Sequences of words
Finite state automata with output
Entropy
Picture languages
Local languages
Dissertations, Academic -- Mathematics -- Masters -- USF ( lcsh )
Genre:
non-fiction ( marcgt )

Notes

Abstract:
ABSTRACT: Transducers are finite state automata with an output. In this thesis, we attempt to classify sequences that can be constructed by iteratively applying a transducer to a given word. We begin exploring this problem by considering sequences of words that can be produced by iterative application of a transducer to a given input word, i.e., identifying sequences of words of the form w, Ï„(w), Ï„²(w), . . . We call such sequences transducer recognizable. Also we introduce the notion of "recognition of a sequence in context", which captures the possibility of concatenating prefix and suffix words to each word in the sequence, so a given sequence of words becomes transducer recognizable. It turns out that all finite and periodic sequences of words of equal length are transducer recognizable. We also show how to construct a deterministic transducer with the least number of states recognizing a given sequence. To each transducer Ï„ we associate a two-dimensional language L²(Ï„) consisting of blocks of symbols in the following way. The first row, w, of each block is in the input language of Ï„, the second row is a word that Ï„ outputs on input w. Inductively, every subsequent row is a word outputted by the transducer when its preceding row is read as an input. We show a relationship of the entropy values of these two-dimensional languages to the entropy values of the one-dimensional languages that appear as input languages for finite state transducers.
Thesis:
Thesis (M.A.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 31 pages.
Statement of Responsibility:
by Egor Dolzhenko.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
001992343 ( ALEPH )
316061185 ( OCLC )
E14-SFE0002380 ( USFLDC DOI )
e14.2380 ( USFLDC Handle )

Postcard Information

Format:
Book

Downloads

This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam 2200397Ka 4500
controlfield tag 001 001992343
005 20090923074702.0
007 cr bnu|||uuuuu
008 090317s2008 flu s 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002380
035
(OCoLC)316061185
040
FHM
c FHM
049
FHMM
066
(S
090
QA36 (Online)
1 100
Dolzhenko, Egor.
0 245
Transducer dynamics
h [electronic resource] /
by Egor Dolzhenko.
260
[Tampa, Fla] :
b University of South Florida,
2008.
500
Title from PDF of title page.
Document formatted into pages; contains 31 pages.
502
Thesis (M.A.)--University of South Florida, 2008.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor: Natasha Jonoska, Ph.D.
3 520
ABSTRACT: Transducers are finite state automata with an output. In this thesis, we attempt to classify sequences that can be constructed by iteratively applying a transducer to a given word. We begin exploring this problem by considering sequences of words that can be produced by iterative application of a transducer to a given input word, i.e., identifying sequences of words of the form w, (w), (w), . We call such sequences transducer recognizable. Also we introduce the notion of "recognition of a sequence in context", which captures the possibility of concatenating prefix and suffix words to each word in the sequence, so a given sequence of words becomes transducer recognizable. It turns out that all finite and periodic sequences of words of equal length are transducer recognizable. We also show how to construct a deterministic transducer with the least number of states recognizing a given sequence. To each transducer we associate a two-dimensional language L() consisting of blocks of symbols in the following way. The first row, w, of each block is in the input language of , the second row is a word that outputs on input w. Inductively, every subsequent row is a word outputted by the transducer when its preceding row is read as an input. We show a relationship of the entropy values of these two-dimensional languages to the entropy values of the one-dimensional languages that appear as input languages for finite state transducers.
653
Sequences of words
Finite state automata with output
Entropy
Picture languages
Local languages
690
Dissertations, Academic
z USF
x Mathematics
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2380



PAGE 1

TransducerdynamicsbyEgorDolzhenkoAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofArtsDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:NatasaJonoska,Ph.D.BrianCurtin,Ph.D.GregoryMcColm,Ph.D.DateofApproval:December14,2007Keywords:SequencesofWords,FiniteStateAutomatawithOutput,Entropy,PictureLanguages,LocalLanguagescCopyright2008,EgorDolzhenko

PAGE 2

AcknowledgmentIwouldliketothankmyadviserDr.NatasaJonoskaforhersupportandmo-tivationeversinceItookmyrstproofbasedmathematicscoursewithher.Also,Iwouldliketothankmycommitteemembers,Dr.BrianCurtinandDr.GregoryMcColmfortheirhelpandusefulcommentstodirectmeinmythesiswork.Lastly,IwouldliketoacknowledgethesupportoftheNationalScienceFounda-tion,asthisworkhasbeensupported,inpart,bytheNSFgrantCCF#0726396.

PAGE 3

ContentsListofguresiiAbstractiii1Introduction11.1Notation.........................................32Recognitionincontext62.1Recognition.......................................62.2Modicationsandfuturedirections........................113Minimization153.1Minimizationofadeterministictransducer...................153.2Minimaltransducerthatrecognizessequenceofwords............174Localpicturelanguages214.1Introduction......................................214.2Notationanddenitions..............................214.3Localpicturelanguages...............................234.4Possibleapplicationsandfuturedirections...................264.5EntropyofL2...................................275Conclusion30References31 i

PAGE 4

Listofgures1Sequenceofperiod8andtransducerrecognizingit.................102TransducerusedinProposition3:3.........................133TransducerthatrecognizessetMwithaip.....................144TransducersuchthatLislocal,butL2isnot...............255Transducergeneratingalocalpicturelanguage....................266Transducergeneratingnon-locallanguage.......................29 ii

PAGE 5

TransducerDynamicsEgorDolzhenkoABSTRACTTransducersarenitestateautomatawithanoutput.Inthisthesis,weattempttoclassifysequencesthatcanbeconstructedbyiterativelyapplyingatransducertoagivenword.Webeginexploringthisproblembyconsideringsequencesofwordsthatcanbeproducedbyiterativeapplicationofatransducertoagiveninputword,i.e.,identifyingsequencesofwordsoftheformw;w;2w;:::.Wecallsuchsequencestransducerrecognizable.Alsoweintroducethenotionofrecognitionofasequenceincontext",whichcapturesthepossibilityofconcatenatingprexandsuxwordstoeachwordinthesequence,soagivensequenceofwordsbecomestransducerrecognizable.Itturnsoutthatallniteandperiodicsequencesofwordsofequallengtharetransducerrecognizable.Wealsoshowhowtoconstructadeterministictransducerwiththeleastnumberofstatesrecognizingagivensequence.Toeachtransducerweassociateatwo-dimensionallanguageL2,consistingofblocksofsymbolsinthefollowingway.Therstrow,w,ofeachblockisintheinputlanguageof,thesecondrowisawordthatoutputsoninputw.Inductively,everysubsequentrowisawordoutputtedbythetransducerwhenitsprecedingrowisreadasaninput.Weshowarelationshipoftheentropyvaluesofthesetwo-dimensionallanguagestotheentropyvaluesoftheone-dimensionallanguagesthatappearasinputlanguagesfornitestatetransducers. iii

PAGE 6

1IntroductionItiscommonlyacknowledgedthatDNAcomputing,asabranchofscience,startedbyAdleman'spaper[1].Sincethen,manyrelatedmodelsofcomputinghavebeendevelopedandexplored.Thesuccessofeachsuchmodeldependsonitscompu-tationalpower,robustness,andcomplexity.SomeofthesemodelsarecloselyrelatedtotheconceptofWangtiles.WecallanitesetofdistinctunitsquareswithcolorededgesasetofWangprototiles.Weassumethateachprototileappearsinanarbitrar-ilylargenumberofcopiescalledtiles.Atilewithleftedgecoloredl,bottomedgecoloredb,topedgecoloredtandrightedgecoloredrisdenotedwith=[l;b;t;r].Norotationorreexionofthetilesisallowed.Twotiles=[l;b;t;r]and0=[l0;b0;t0;r0]canbeplacednexttoeachother,totheleftof0ir=l0,and0ontopofit=b0.MoreinformationaboutWangtilescanbefoundin[6].Recently,aphysicalrepresentationofWangtileswithDNAmoleculeshasbeendemonstrated[13,14].Itiswellknownthatbyiterationofgeneralizedsequentialmachinesnitestatemachinesmappingsymbolsintostringsallcomputablefunctionscanbesimulatedseeforex.[9,10].Thefullcomputationalpowerdependsonthepossibilityforit-erationsofanitestatemachine.AsthereisanaturalsimulationoftheprocessofiterationoftransducersandrecursivecomputablefunctionswithWangtiles[7],thisideahasbeendevelopedfurtherin[3]whereasuccessfulexperimentalsimulationofaprogrammabletransducernitestatemachinemappingsymbolsintosymbolswithDNAWangtileshavingiterationcapabilitiesisreported.Thisexperimentaldevelop-mentprovidesmeansforgeneratingpatternsandvarietyoftwo-dimensionalarraysatthenanolevel.WegiveabriefexamplebyillustratingconnectionofthetransducerstoWangtilescompletedescriptionofthismodelcanbefoundin[7].Considertransducerpicturedbelow. 1

PAGE 7

0 0 0 1 0 1 q0 q1 ToeachtransitionofthistransducerwecanassociateWangprototilesasfollows. q0 q0 0 0 0 q0 0 q0 q1 q1 1 1 Whereeachstateandeachsymbolofthealphabetrepresentsadistinctcolor.Thesetilescanbeassembledintotherowdepictedbelow. 0 1 q1 q0 0 0 q0 q0 0 1 q0 q1 0 0 q0 q0 Noticethatthebottomedgeofthisrowrepresentsinputword0100andtopedgerepresents0010,thatis,0010istheoutputofthetransducerontheinput0100.Sim-ilarlywecanconstructanotherrowoffourtiles,withbottomedgerepresentingword0010andthetopedgeword0001bystackingthisrowontopoftherst.Continuinginthiswaywecanconstructablockofarbitraryheight.Theaboveexampleillustratesthemaingoalofthiswork,theclassicationofpatternsthatcanbegeneratedbythedescribedprocess.Webeginexploringthisproblembyconsideringsequencesofwordsthatcanbeproducedbyiterativeappli-cationofatransducertoagiveninputword,i.e.,identifyingsequencesofwordsoftheformw;w;2w;:::.Wecallsuchsequencestransducerrecognizable.Alsoweintroducenotionofrecognitionofasequenceincontext",whichgivesrisetoapossibilityofconcatenatingprexandsuxtoeachwordinthesequence,soagivensequenceofwordsbecomestransducerrecognizable.Itturnsoutthatallniteandperiodicsequencesofwordsofequallengtharetransducerrecognizable.Additionallywebrieyexploreotherwaystoiterativelyapplyatransducertoawordinordertogetasequence.Thenextquestionwhichweconsideristhefollowing:Givenasequenceofwordss,howcanoneconstructadeterministictransducerwiththeleastnumberofstates 2

PAGE 8

recognizings.First,weconrmthatafterminormodicationswecanapplyanalgorithmforminimizationofMealymachinestodeterministictransducers.Henceifweapplythisalgorithmtoanytransducerweobtainanequivalenttransducerwiththesmallestpossiblenumberofstates.Byequivalenttransducerswemeantransducersthatac-ceptthesamesetofwords,andforeachacceptedword,bothtransducersproducethesameoutput.Ingeneral,however,applicationofthisalgorithmtoatransducerthatrecognizessequencesdoesnotresultinatransducerwiththesmallestnumberofstatesthatrecognizessequences.Asalreadynoted,theminimizationalgorithmalwaysresultsinatransducerequivalenttotheoriginalone,andtwotransducersrec-ognizingsequencesdonothavetobeequivalentsincetheycandieronthewordsthatarenotpartofthesequences.Toovercomethisproblemwedenearelationonthesetofstatesthatindicatesthestatesthatcanbeinsomesensejoinedtogetherintoonestate,suchthattheresultingtransducerremainsdeterministicandrecognizessequences.Weshowthatthisrelationprovidesawayforatransducerwithminimalnumberofstatestobeconstructed.Nextwenotethateverynitetransducerrecognizablesequencew;w;2w;:::canbeassociatedwithatwo-dimensionalblockwhoserstrowisw,secondrowisw,andithrowisiw.Foragiventransducer,wedenoteallpossibletwo-dimensionalblocksthatcanbeconstructedinthiswaybyL2.WeanalyzetheconnectionbetweenL2andtheinputlanguageof,L,fromthepointofviewoflocallanguagesandentropy.WeobservethattheentropyL2fordeterministictransducerisalwayszero.Howeverifweconsidernondeterministictransducers,i.e.transducersthatcanhavemorethanoneoutputforaninputword,thisisnolongertrue.Infactitturnsoutthatforanygivennondeterministictransducer,theentropyofLisalwaysanupperboundfortheentropyofL2.Notethatpartsofthisthesishavebeensubmittedforpublication[5].1.1NotationAnonemptynitesetAiscalledanalphabet.MembersofthealphabetAarecalledsymbols.AwordoveralphabetAisanitesequenceofsymbolsfromA,whoseelementsarewrittennexttooneanotherandnotseparatedbycommas.Thelength 3

PAGE 9

ofawordwisanumberofsymbolsinwandisdenotedbyjwj.Awordoflengthzeroisdenotedby.LetAdenotethesetofallpossiblewordsoveralphabetAandletA+=Anfg.Forawordwletwideneithsymbolofthisword.Alsoifw=ww:::wn)]TJ/F15 11.955 Tf 12.122 0 Td[(1wnthenwR=wnwn)]TJ/F15 11.955 Tf 12.122 0 Td[(1:::ww.Forexample,ifw=011thenw=1andwR=110. Denition1.1. ADeterministictransducerisasix-tuple=A;Q;;;q0;F;whereAisanitealphabet,Qisanitesetofstates,q0isaninitialstateq02Q,FisasetofnalstatesFQ,isatransitionfunction:QA!Q,andisanoutputfunction:QA!A.Transducersareoftenrepresented,andevendened,bydiagrams.Givenadeterministictransducer=A;Q;;;q0;FwestartconstructingitsdiagrambydepictingthenamesofthestatesinQ.Next,foreachpairq1;a;q2inandq1;b;q2inwedrawanarrowfromq1toq2.Abovethisarrowweputsymbolb a,andrefertoaasaninputlabelandtobasanoutputlabelofthisarrow.Weindicatethatq0isaninitialstatebythesmallarrowpointingatit.Finallywecirclenalstates,i.e.allthemembersofthesetF.Sincemostofthisworkdealswiththedeterministictransducers,wewillrefertodeterministictransducerssimplyastransducers. Denition1.2. For=A;Q;;;q0;F,q2Q,anda2A,letq;a=q;a.Forw=av;a2A,andv2A+,letq;w=q;a;v. Denition1.3. For=A;Q;;;q0;F,q2Q,anda2A,letq;a=q;a.Forw=av;a2A,andv2A+,letq;w=q;aq;a;v.Fromnowonreferstoandrefersto.SincecoincideswithandcoincideswithonA,thisshouldnotproduceanyambiguities. Denition1.4. Awordw2Aisacceptedbyatransducer=A;Q;;;q0;Fifq0;w2F.ThesetofallwordsthatacceptsisdenotedL.IfthestateqinQhasthepropertythatforallwordsw,q;w=2Fthensuchqisdenotedasqjunk. 4

PAGE 10

Weassumethatanytransducercontainsatmostonestateqjunk. Denition1.5. Foratransducer=A;Q;;;q0;Fandw2Aifacceptswandq0;w=u,thenwewritew=u.LetRngdenotethesetofallwordsusuchthatthereisw,suchthatw=u. Denition1.6. Anondeterministictransducerisave-tuple=A;Q;;q0;FwhereAisanitealphabet,Qisanitesetofstates,q0isaninitialstateq02Q,FisasetofnalstatesFQandisatransitionrelationQAAQ.Notethatalloftheabovenotionsaredenedsimilarlyfornondeterministictransducers,however,inthedenitionofthenondeterministictransducer,isnotafunction.Forinstance,givenwinL,wnowdenesaset,sincemayhavemorethanoneoutputonw. 5

PAGE 11

2RecognitionincontextIn[2]theauthorsdescribeawaytouseWangtilestosimulateatransduceronaninputwordw.Briey,eachtilerepresentsaspecictransition,withcolorsoftheleftandrightedgesencodingsourceandtargetstates,andtopandbottomcolorsencodinginputandoutputsymbols.Thereisarowofjwjtileswiththebottomedgeencodingwordw,suchthatleftmostverticaledgeencodesq0,initialstateof,therightmostverticaledgeofthisrowisoneoftheterminalstatesof,andalloftheadjacentverticaledgesencodethesamestate.Thisimpliesthattopedgeofthisrowencodesw,theoutputofoninputw.Notethatwhenisdeterministic,therowwiththepropertiesaboveisunique.Incasethatwordwisacceptedby,thereisanotherrowwiththepropertiesmentionedabove,suchthatitstopedgeiswandthebottomedgeisw.Continuinginthiswayandplacingtheserowsontopofoneanothertwodimensionalblockisobtained,correspondingtothesequencew,w,2w;:::.Themaingoalofthissectionistoformalizetheabovediscussionthroughthenotionsofrecognitionandrecognitionincontextandtodiscusswhatsequencesofwhatperiodscanbeobtainedthroughsuchprocess.Furthermore,lastsectiondiscussessetsofsequencessuchthateachsequenceinthesetisoftheforms1,s1,2s1;:::,wheres1denotesitsrstelement.Forexamplethismaybethecasewhenallsequenceshavesimilarstructureanddieronlyinthelengthoftheirwords.2.1RecognitionThissectiondealsonlywithdeterministictransducers. Denition2.1. Ifs=s1;s2;:::;sk,isanitesequenceofwords,thenwewrite#s=kandifsisinnite,then#s=1. Denition2.2. Asequences=s1;s2;:::iscalledperiodicifthereexistsp2Nsuchthatforallsi=sp+i,i2N.Theleastsuchpiscalledtheperiodofs. 6

PAGE 12

Denition2.3. Lets=s1;s2;s3;:::beasequenceofwordsoveralphabetAsuchthatjsij=jsjjforalli;j.Ifthereexistsadeterministictransducersuchthatsi+1=sifor1i<#s,thensissaidtobetransducerrecognizableandissaidtorecognizes. Denition2.4. Atransducerrecognizespreciselyasequencesifrecognizessandforanyothersequencet,suchthatrecognizest,thereisanaturalnumbern,suchthatns1=t1.Heres1andt1denoterstelementsofsequencessandtrespectively. Proposition2.1. Lets=s1;s2;:::beasequenceofwordsoverAwithjsij=jsjj=kforall1i;j<#s.Thenthissequenceistransducerrecognizableifandonlyiffollowingholds:Forallr=1;:::;k;if8t=1;:::;r,sit=sjtthensi+1r=sj+1rforalli;j. Proof. Incaseholds,considerthetransducernotethatdenotesanemptyword=A;Q;;;;Fthatwouldrecognizes,whereQ,thesetofstates,isdenedbyQ=fg[fsi:::sitjsi2sand1tkg:sincekisxed,thissetisnite.F=fsijsiisamemberofasequencesgForeverysi2ssuchthat1i<#sdenesi:::sit)]TJ/F15 11.955 Tf 11.956 0 Td[(1;sit=si:::sitandsi:::sit)]TJ/F15 11.955 Tf 11.955 0 Td[(1;sit=si+1t: 7

PAGE 13

Also,add;si=siand;si=si+1.Thentheoutputsymbolisuniquelydeterminedbysi:::sitdueto.Henceisdeterministic.FinallyaddqjunktoQandletallthemissingtransitionsleadtoit.Notethatq0;w2F,i.e.,thistransduceracceptswordwifandonlyifw2sandthat;si=si+1byconstruction.Thusrecognizess.Conversely,ifforsomesi;sj2swith1i;j<#sandforsomelhavesi:::sil=sj:::sjlbutsi+1l6=sj+1landthereisdeterministictrans-ducer=A;Q;;;q0;Fthatrecognizessthenletq=q0;si:::sil)]TJ/F15 11.955 Tf 12.895 0 Td[(1=q0;sj:::sjl)]TJ/F15 11.955 Tf 13.554 0 Td[(1.Thisisacontradiction,sincesi+1l=q;sil6=q;sjl=sj+1l. Ifsisatransducerrecognizablesequence,letsdenotethetransducercon-structedbythealgorithmintheproposition2.1thatrecognizess.Notesomeofthepropertiesofs: i. Ifq0;w2F,thenbyconstructionofs,q0;w=wandw2s ii. Ifq0;w0=q0;w00,thenw0=w00,sincebyi,w0=q0;w0=q0;w00=w00. iii. Thetransducersrecognizessprecisely,sincerecognizessandforanyothersequence,w;w;2w:::,thatrecognizes,wmustbeacceptedbyandthen,byi,w2s.SupposejAj=2andletsbeatransducerrecognizableperiodicsequenceoveralphabetAofperiodgreaterthanone.Thentheperiodofsmustbeeven.ToseethissupposeA=fa;bg.Lettbetheleastnaturalnumbersuchthatthereisiandj,sit6=sjt.Thenitfollowsthats1ts2ts3t:::=ababab:::ors1ts2ts3t:::=bababa:::.Thisisso,sinceduetodeterminismof,itmustbetruethatq0;si:::sit)]TJ/F15 11.955 Tf 12.076 0 Td[(1=q0;sj:::sjt)]TJ/F15 11.955 Tf 12.075 0 Td[(1=:q0foralli;j.First,q0;a=aandq0;b=bcannothappenbythechoiceoft,sinceaboveimpliesthatsit=s1tforalli.Second,q0;a=aandq0;b=acan'thappeneither,sinceinthiscasesit=aforalliors1t=bandsit=a,i>1inwhichcasethesequenceisnotevenperiodic.Thusoneofthetwocasesmentionedabovemustbetrue,whichimpliesthatperiodofthesequencesmustbeeven. 8

PAGE 14

Proposition2.2. Foreachnaturalnumbern,thereexistsatransducerrecognizablesequencefsig10overalphabetA=fa0;a1;:::;akgofperiodjAjn. Proof. byinductiononnForn=1,letsi=aimodjAj.Sincesi=sjimpliesthataimodjAj=ajmodjAjandijmodjAjwehavethati+1j+1modjAjandsi+1=sj+1.Thusthissequencesatisestheconditionsoftheorem2.1,andthusitistransducerrecognizableandclearlyofperiodjAj.Assumethatthepremiseholdsforn=t)]TJ/F15 11.955 Tf 11.955 0 Td[(1,i.e.thatthereexistsasequencefs0ig10ofperiodjAjt)]TJ/F19 7.97 Tf 6.587 0 Td[(1.Forn=jAjtletsi=s0iammodjAjsuchthati=mjAjt)]TJ/F19 7.97 Tf 6.586 0 Td[(1+rwhere0r
PAGE 15

q0 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 011 101 001 110 010 100 000 111 Figure1:Sequenceofperiod8andtransducerrecognizingit denenewsequencefsigk)]TJ/F19 7.97 Tf 6.586 0 Td[(10bypi=bi,i=0:::k)]TJ/F15 11.955 Tf 12.619 0 Td[(1.Thenfwigisrecognizableincontextwithprexfpig.Toseethisnotethatifpiwi=pjwjthenpi=pj,hencei=j.Inotherwordssequencefpiwigk1sucesthepremiseoftheProposition2.1. Corollary2.2. EverysequenceconsistingofwordsofequallengthofperiodjAjnisrecognizableincontext. Proof. LetfwigbethesequenceofperiodjAjn.Theorem2.2yieldstransducerrec-ognizablesequencesequencefpigofperiodjAjn.Hence,similarlytothepreviouscorollary,sequencefwgisrecognizableincontextwithprexfpig. Denition2.6. LetD=fs1;s2;s3;:::gbeasetofsequencesofwordsofequallengthoveralphabetA.Thenifthereexistsadeterministictransducer,suchthatrecognizessiincontext,foralli=1;:::;jDj,withthesameprexandsuxforeverysetsi,thenwesaythatrecognizessetDandthatsetDisrecognizableincontext.Thiswaywerequireofonetransducertorecognizeasetofsequenceswiththesameprexandsux. Example2.1. SetM=ff0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gni=1jn=2;3;:::gisnotrecognizableincontext. 10

PAGE 16

Proof. SupposethatthereexistssuchthatrecognizesMincontextwithprexpandsuxd.Considertherstelementofthesequencef0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.587 0 Td[(1gni=12M,suchthatn)]TJ/F15 11.955 Tf 11.254 0 Td[(1isgreaterthantwicethenumberofstatesof.Thenitfollowsthatonthesubword0n)]TJ/F19 7.97 Tf 6.587 0 Td[(11ofp10n)]TJ/F19 7.97 Tf 6.586 0 Td[(11d1transducergoesthroughstatesqt1;qt2;qt3;:::;qtn+1andyields0n)]TJ/F19 7.97 Tf 6.587 0 Td[(210,i.e., qt1;0=;qt2,qt2;0=;qt3,:::qtn)]TJ/F20 5.978 Tf 5.756 0 Td[(1;0=1;qtn;qtn;1=0;qtn+1:Bythepigeonholeprinciple,therearetm;tlsuchthatqtm=qtlandtm
PAGE 17

Denition2.7. Lets=s1;s2;s3;:::beasequenceofwordsoveralphabetAsuchthatforalli;jjsij=jsjj.Ifthereexistsadeterministictransducerandsequencep=p1;p2;p3;:::suchthat8i;jjpij=jpjj,andsequenced=d1;d2;d3;:::with8i;jjdij=jdjjand#d=#p=#ssuchthatpi+1si+1di+1=pisidiRRfori=1;2;3;:::;#s)]TJ/F15 11.955 Tf 11.899 0 Td[(1.Thenissaidtorecognizeswithaipincontextandsisrecognizablewithaipincontext.Andhencewecancorrespondinglyadjustdenitionforrecognitionofaset. Denition2.8. LetD=fs1;s2;s3;:::gbeasetofsequencesofwordsoveralphabetA.Thenifthereexistsadeterministictransducersuchthatrecognizessiwithaipincontextforalli=1:::jDj,withthesameprexandsuxforeverysetsi,thenwesaythatrecognizessetDwithaipincontextandthatsetDisrecognizablewithaipincontext. Proposition2.3. Alltransducerrecognizablesequencesarerecognizablewithaipincontext. Proof. Lets=s1;s2;s3:::beatransducerrecognizablesequenceofwordsoverA.Let=A;Q;;;q0;Fbethetransducerthatrecognizess.Usingletsdene0=A;Q0;0;0;q00;f0asitisdoneinFigure2.2.Letprexpandsuxdbedenedbypi=0anddi=1foralli.Thisway0willbeabletodistinguishbetweenwandwRforeachw2s.Since00pisidiRR=00si1RR=00sRi0R=0sRi0R=0si1=0si+11,itfollowsthatsisrecognizablewithaipincontext. Example2.2. M=ff0n)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gni=1jn=2;3;:::gisrecognizablewithaipincon-text. Proof. Lets2M.Thuss=f0t)]TJ/F22 7.97 Tf 6.586 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1gti=1forsomet.Thensi=0t)]TJ/F22 7.97 Tf 6.587 0 Td[(i10i)]TJ/F19 7.97 Tf 6.586 0 Td[(1istheithwordinthesequences.Deneprexpandsuxdbypi=1anddi=0foreachi.LettransducerbeasitisdenedintheFigure2.2.Then 12

PAGE 18

q00 ::: q01 q02 q0k qt1 q0 qt2 q0k+2 q0k+1 Figure2:TransducerusedinProposition3:3 q5 q6 q0 q3 1 q4 q1 1 1 0 1 0 0 0 0 1 1 1 0 q2 0 0 0 0 q7 1 1 1 0 0 Figure3:TransducerthatrecognizessetMwithaip wRiR=0t)]TJ/F22 7.97 Tf 6.587 0 Td[(i10i)]TJ/F19 7.97 Tf 6.587 0 Td[(10RR=0i)]TJ/F19 7.97 Tf 6.587 0 Td[(110t)]TJ/F22 7.97 Tf 6.586 0 Td[(i1R=0i10t)]TJ/F22 7.97 Tf 6.586 0 Td[(i)]TJ/F19 7.97 Tf 6.587 0 Td[(11R=0t)]TJ/F22 7.97 Tf 6.586 0 Td[(i)]TJ/F19 7.97 Tf 6.587 0 Td[(110i1=10t)]TJ/F22 7.97 Tf 6.587 0 Td[(i)]TJ/F19 7.97 Tf 6.586 0 Td[(110i0=1si+10.ThussetMisrecognizablewithaipincontext. 13

PAGE 19

3MinimizationAMealymachineisavetuple=A;Q;;;q0whoseelementshavethesamedenitionastherstveelementsofthedeterministictransducer.Hencede-terministictransduceraresomewhatmoregeneralthanMealymachines,whichhavebeenstudiedwell.Howeveraimofthefollowingsectionistoshowthatwhenitcomestominimizationthesamealgorithmcanbeapplied.Thesecondsectiondealswithndinginsomesensesmallestdeterministictransducer,ifoneexists,thatcanrecognizegivensequence.3.1MinimizationofaDeterministicTransducerThedenitionofthedeterministictransducerdescribedaboveismoregeneralthanthatoftheMealymachine,however,onlydierenceisthepresenceofthenalstatesindeterministictransducer,whichcanbeconrmedin[4].Thusitseemsplausibletousealgorithm,similartothealgorithmfortheminimizationofaMealymachine,thatwouldtakeintoaccountthesetofthenalstates.Hencethecontentofthissectionis,althoughmodiedformoregeneralcase,takenfrom[4]. Denition3.1. Astateaoftransducer=A;Q;;;q0;Fisaccessibleifthereissomeinputwordw2Asuchthatq0;w=q.Transducerisconnectedifeverystateisaccessible.Forsimplicityassumethatalltransitionsgoingtothestateqjunkhaveinputsymbolequaltotheoutputsymbol,i.e.q;a=a;qjunk,whereqjunkisthestateforwhich8w2Aqjunk;w=2F.Also,withoutlossofgenerality,itisassumedthatqjunkistheonlystatewiththispropertyandthateverytransducerisconnected. Denition3.2. Foratransducerandw2A,letw=uifacceptswandoutputsu. Denition3.3. Twotransducers1=A1;Q1;1;1;q01;F1 14

PAGE 20

and2=A2;Q2;2;2;q02;F2areequivalentifandonlyif 1. A1=A2,and 2. L1=L2andforallw2L1w=2w.Asarelation,equivalenceestablishesanequivalencerelationonasetofalldeterministictransducers. Denition3.4. Twostatesqaandqboftransducer=A;Q;;;q0;Fareequiv-alentifandonlyifa=A;Q;;;qa;Fisequivalenttob=A;Q;;;qb;F. Proposition3.1. Twostatesqaandqbofatransducer=A;Q;;;q0;Fareequivalentifandonlyif 1. 8s2Aqa;s=qb;s,and 2. 8s2Aqa;sisequivalenttoqb;s. Proof. Ifbothoftheconditionshold,thenletw2Aandw=snwheres2A,thent=qa;s=qb;sandqa;sn=qb;sn=k.Thusqaw=qbw=tk.Conversely,letqabeequivalenttoqb,thenletw=sn,wheres2Aandn2A.Sinceasn=bsnitfollowsthatqa;sn=qb;snhenceqa;sisequivalenttoqb;s.Alsoqa;s=qb;ssinceifqa;s=2qjunkthereexistsw0suchthatqa;sacceptsw0andhenceqaacceptssw0,implyingthatqasw0=qbsw0,thusqa;s=qb;s.Ifqa;s=qjunkandqb;s=qjunkthenqa;s=qb;s=sbypreviousassumptionthattransitionsleadingtoqjunkproduceoutputequaltoinput. Denition3.5. Atransducerisreducedifitcontainsnopairofequivalentstates. 15

PAGE 21

Denition3.6. Statesqaandqbofatransducer=A;Q;;;q0;Farek-distinguishableifthereexistsawordw2Ajwjk,suchthatqaw6=qbw:Thenwiscalledadistinguishingword. Denition3.7. Iftwostatesqaandqbarenotkdistinguishable,thentheyarek-equivalent. Proposition3.2. Twostatesqaandqbofatransducerarek-equivalentifandonlyif 1. Theyare1-equivalent. 2. Foreachs2Aqa;sandqb;sarek)]TJ/F15 11.955 Tf 11.955 0 Td[(1equivalent. Proof. Essentiallythesameasinpreviousproposition. Iftwostatesofatransducerarek-equivalentforallkthentheyareequivalent.Thisrelationdenesapartitionofthesetofthestatesofatransducer,whichisusedforconstructionofanew,reducedtransducer.Thealgorithmgivenin[4]couldbeusedinspitethefactthatitwaswrittenforMealymachines,aslongasappropriatedenitionsareused.3.2MinimalTransducerthatRecognizesSequenceofWordsIntheprevioussectionwetriedtominimizethenumberofstatesinthedeter-ministictransducer.Forinstancegivenatransducerrecognizablesequenceofwordssthealgorithmin[4]canbeappliedtostondanequivalenttransducer,butwiththeminimalnumberofstates.Ontheotherhandif,foragivensequences,weneedtondatransducerwiththeminimalnumberofstatesthatwouldrecognizes,theabovealgorithmwouldnotwork,astheremaybeothertransducers,notequivalenttosthatalsorecognizes.Probablythemostbasicalgorithmforndingtheminimaltransducerthatwouldrecognizeagivensequencescouldproceedasfollows: 1. ForagivensequencesconstructsusingalgorithmoutlinedinProposition2.1. 16

PAGE 22

2. Sincethereareonlynitelymanytransducersovernitealphabetwithxednumberofstates,enumeratealltransducersthathavefewerstatesthanthetransducerconstructedinstep1. 3. SincescannotbeaperiodictobetransducerrecognizableandthereareonlyjAjkdierentwordsoflengthkoveralphabetA,atransducerwiththeminimalnumberofstatesrecognizingscanbefoundinnitelymanysteps.Intheprevioussection,anequivalencerelationonthesetofstateswasusedtodeterminethestatesthatareequivalentandanewtransducer-obtainedthroughequivalenceclassesofthatrelation-wasequivalenttotheoriginalone.Here,aslightlydierentapproachmustbetaken.Theideaistoexamineanewrelationonthesetofstatesthatwouldindicatethestatesthatinsomesensecanbejoinedtogetherwithoutaectingthetransducer'sabilitytorecognizeagivensequence.Let0denotethetransducerconstructedfrombymakingallofthestatesof,exceptqjunk,nal.Thenletdomdenoteallofthewordsacceptedby0. Denition3.8. Letq1;q22Q,thenstatesq1andq2areinrelationq1q2ifandonlyifforallw2domq1domq2havethatq1;w=q2;w. Proposition3.3. Ifq1q2ands2domq1domq2Athenq1;sq2;s. Proof. Letq1,q2,sbeasdescribedinpremiseand:q1;sq2;s.Thenitfollowsthat9w2domq1;sdomq2;s,suchthatq1;s;w6=q2;s;w.Thensw2domq1domq2andq1;sw=q1;sq1;s;w6=q2;sq2;s;w=q2;sw,whichisacontradiction. Proposition3.4. Letbeadeterministictransducerrecognizingthesequences=s1;s2;:::,andletEfu;vju;v2Qanduvgbesuchthat: EdenesanequivalencerelationonsetofstatesQ 8u;v2E,ifa2domudomvA,thenu;a;v;a2E.Thenthereexists0,withsetofstatesequaltothesetoftheequivalenceclassesproducedbyE,suchthat0recognizess. 17

PAGE 23

Proof. LetQ0denotethesetofequivalenceclassesinducedbyE.Denotetheequiva-lenceclasstowhichqbelongsby[q].Then8[q]2Q0and8a2A,letq02[q]beastatesuchthatq0;a6=qjunk.Nowdene0[q];a=[q0;a]and0[q];a=q0;a.If8p2[q],p;a=qjunkthen0[q];a=[qjunk]and0[q];a=a.Duetotheassumptions,ifq1;q22[q],andq1;s6=qjunkandq2;s6=qjunk,i.e.,s2domq1domq2A,then[q1;a]=[q2;a]andthustheabovecon-structionproducesadeterministictransducer0=A;Q0;0;0;[q0];Q0withoutanyambiguities.Weshowthat0recognizessequences.Letskandsk+1betwoconsecutivewordsofthesequences.Letgothroughthestatesq0;q1;:::;qnoninputsk,whereqi;ski+16=qjunk.Correspondingly,thetransducer0willgothroughthese-quenceofstates[q0];[q1];:::;[qn]andproducesk+1,sinceforanyi=0:::n)]TJ/F15 11.955 Tf 13.254 0 Td[(1qi;ski+1=qi+1impliesthat0[qi];ski+1=[qi+1]and0[qi];ski+1=qi;ski+1=sk+1i+1.Thusthistransducerrecognizess=s1;s2;:::andthesizeofjQjisequaltothenumberofequivalenceclassesofjEj. LetEbearelationonthesetofstatesofthedeterministictransducerthatsatisesthepremiseoftheProposition3.4.WesaythatEyields0if0isconstructedthroughthealgorithmoutlinedintheProposition3.4usingrelationE. Proposition3.5. LetsbethesequenceofwordsoverAandletsbethetransducerconstructedbythealgorithminProposition2.1thatrecognizess.LetEbeasetthatsatisestheconditionsofProposition3.4,thatwouldproducethesmallestnumberofequivalenceclasses.ThenEyieldsatransducerwithminimalnumberofstatesthatrecognizesthesequences. Proof. Lets=A;Q;;;q0;F,andlet0=A;Q0;0;0;q00;F0denoteaminimaltransducerwiththeminimalnumberofstatesthatwouldrecognizes.Considerfollowingrelation:pqifandonlyifp;q6=qjunkandtherearewordsn1;n22Asuchthatq0;n1=pandq0;n2=qand0q00;n1=0q00;n2.Wesaythatwordsn1andn2correspondtostatespandq. 1. RelationisanequivalencerelationonQnfqjunkg.Itisclearthatitisreexive 18

PAGE 24

andsymmetric.Alsoifq1q2andq2q3then9n01andn02correspondingtoq1andq2andn001andn002correspondingtoq2andq3.Duetothepropertiesofs,itmustbethatn02=n001.Hence0q00;n01=0q00;n02=0q00;n001=0q00;n002.Thusq1q3. 2. ifpqthenifforsomen1;n22Aq0;n1=pandq0;n2=qand0q00;n1=0q00;n2hence0q00;n1a=0q00;n2aandp;aq;a8a2dompdomqA. 3. Ifpqthentherearen1andn2asdescribedabove.Letw2dompdomqhencep;w6=qjunkandq;w6=qjunk.Fromthedenitionofsitfollowsthat9n01;n022Asuchthatn1wn012sandn2wn022s.Itfollowsthat00q00;n1;w=00q00;n2;w.Sinceoutputsofbothtransducersmustagreeonn1wn01andn2wn02itfollowsthatp;w=q;w.Thuspq.Sinceforeachp2Qnfqjunkg,p2Aandq0;p=p,andthewordpwiththispropertyisuniqueitfollowsthattheequivalenceclassesofcanbeputinonetoonecorrespondencewiththesubsetofstatesof0asfollows:Leteach[q]correspondto0q00;w,wherewissuchthatq0;w=q.NowconstructanequivalencerelationEbyaddingqjunkintoanyoneoftheequivalenceclassesoftherelation.ItfollowsthatEsatisesthepremiseoftheProposition3.4.ThusEyieldstransducerwithminimalnumberofstatesthatrecog-nizess. Thusitwasshownthat,ifforagivensequences,sisatransducerthatrecog-nizespreciselyswithpropertiesgiveninproposition2.1thenitispossibletoconstructaminimaltransducerthroughrelationsimilarlyasitwasdoneintherstsection.Themaindierenceisthatinthiscasethingsarealittlemorecomplicatedsinceisnotanequivalencerelation. 19

PAGE 25

4Localpicturelanguages.4.1IntroductionLetLdenotetheinputlanguageofthetransducerandletL2denotethetwo-dimensionallanguageassociatedwithiterativeapplicationsofagiventransducertowordsofL.InthissectionwewillattempttoanalyzetherelationbetweenL2andLfromthepointofviewoflocallanguagesandentropy.Weobservethatfordeterministictransducers,entropyisalwaysequaltozero,howeverthisisnotthecaseifwewillconsiderthetwo-dimensionallanguagecorrespondingtoanondeterministictransducers.4.2Notationanddenitions Denition4.1. ApicturelanguageoveralphabetAisasubsetofAwhereAdenotesthesetofallpossiblerectangularblocksoveralphabetA Denition4.2. AlocalpicturelanguageLoforderkisapicturelanguagesatisfyingB2LifandonlyifFk;kBQk;k,whereQk;kisanitesetofkkblocksandFk;kBdenotesthesetofallkksubblocksofB.Thenotationusedinthissectionisillustratedusingthefollowinggure: k l i j 20

PAGE 26

TheshadedverticalsubblockofthedepictedblockcanbedescribedasB[][i:::j],whileB[k:::l][]standsforashadedhorizontalblock.TheintersectionofB[k:::l][]andB[][i:::j]isdescribedbyB[k:::l][i:::j].TheithcolumnofBisdenotedasB[][i],andkthrowisB[k][].Notethatthisnotationisalsoapplicableto1-dimensionalwords,sincetheycanbeconsideredasblocksofunitheight.IfB1,B22AsuchthatB1isablockofsizemnandB2istheblockofsizemk,thenwedenetheconcatenationofB1andB2tobetheblockCofsizemn+ksuchthatC[][1:::n]=B1andC[][n+1:::n+k]=B2.Uptothelastsectionofthischapteronlydeterministictransducerswereconsid-ered,inwhichcaseweusethefollowingconvention:Foratransducerandw2L,letnwdenoteanjwjblockB2AwithB[1][]=wandB[k][]=B[k)]TJ/F15 11.955 Tf 12.006 0 Td[(1][]for1
PAGE 27

4.3LocalpicturelanguagesIngeneral,ifatransducerhasalocallanguageasitsinputlanguagethenL2doesnothavetobelocal.Forexample,considerthetransducerdepictedinFigure4.Thistransducerisdeterministicbothoninputandoutput.Allofitsstatesarenal,andoutputsymbolsaredepictedinsidethestates,sincetheyarethesameforallofthetransitions.Theinputlanguageofthetransducer,Lislocalnobbb.SupposeL2islocaland,asinthedenition,thereisasetQk;kofallowedblocksofsizekkandA;B;C2Aaredenedasfollows:A=:::aaaa:::aaaaaaa:::aaaabaa:::aabB=:::aaaa:::aaaaaaa:::aabbaaa:::abaC=:::aaaa:::aaaaaaa:::aababaa:::abaNotethattheblocksA;B2L2canbeextendedindenitelyinheightandwidth.ThusifAandBareextendedtothekkblocksthenA;B2Qk;k.AlsoCcouldbeextendedtohavelengthofk+1,inwhichcasewegetthatFk;kC=fA;BgQk;kThusitmustbethatC2L2.However,thisisacontradiction,sinceoninputC[1][]bottomrowofblockCtheoutputofisdierentfromC[2][].Hencethelanguageisnotlocal.SupposethatL2isalocalpicturelanguage.DoesthisimplythatLis 22

PAGE 28

Figure4:TransducersuchthatLisalocal,butL2isnot q3 a a b a b a q0 q1 Figure5:Transducergeneratingalocalpicturelanguage local?IfL2islocalwithQk;kthentherstthingthatcomestomindistoconsideralocallanguagewithsetofallowedwordsequaltothesetofallwordsinLoflengthk.HoweveritmayhappenthatthereareblocksB1andB2inQk;kwithrstrowsw1andw2respectively,suchthatw1andw2overlap:w1[][2:::k]=w2[][1:::k)]TJ/F15 11.955 Tf 12.041 0 Td[(1],butB1andB2donot,i.e.,B1[][2:::k]6=B2[][1:::k)]TJ/F15 11.955 Tf 11.955 0 Td[(1].Example:LetB1=abaaandB2=abbaandbethetransducerdepictedonFigure5.NotethatL2islocalalmostlocal.Infact,L2nfB2AjF2;2B=fgg=fB1;B2g.Also,B1andB2donotoverlap,butw1=ba,w2=aado.Astheaboveexampleshows,weneedtoenlargethesetofallowedwordsby 23

PAGE 29

includingwordsinLoflengthk+1.Thisway,ifthetwowordsoflengthkoverlapasdidw1andw2above,theirresultingword,i.ew=w1twheret=w2[][k]willbepresentinthelocallanguageonlyifw2L,which,sinceL2islocalpicturelanguage,makessurethatcorrespondingB1andB2alsooverlap.Moreformallywehavethefollowingproposition: Proposition4.1. LetM=fw2Ljkwisundenedg.IfL2isalocalpicturelanguageoforderk,thenLnMmustbealocallanguagewiththesetofallowedwordsP=fwjw2LnMandjwj=korjwj=k+1g. Proof. LetHAbealocallanguagewithPasthesetofallowedwords.Ifw2Hwherejwj=k,thenw2LnMbydenitionofH.Ifjwj>kthenconsiderthefollowingconstruction:B=kw[1:::k]kw[2:::k+1][][1]:::kw[jwj)]TJ/F21 11.955 Tf 17.933 0 Td[(k:::jwj][][1]Sincew[1:::k+1]2LnM,duetotheassumptionskw[1:::k+1]isdened.SinceL2islocalpicturelanguage,Fk;kkw[1:::k+1]=fkw[1:::k];kw[2:::k+1]g:ThusB=kw[1:::k+1]:::kw[jwj)]TJ/F21 11.955 Tf 17.849 0 Td[(k:::jwj][][1].ContinuinginthiswaygetthatB=kw.Ifw2LnM,thenkwisdened.Henceifuisanyfactorofw,i.e.Fk;kkw[jvj:::jvuj]Fk;kkwQk;k:Thusifjuj=kthenu2LnMandifjuj=k+1,u2LnM.Hencew2H. Corollary4.1. IfL2isalocalpicturelanguageandLLthenLislocal. Proof. Sincekwisdenedforeveryw,M=fw2Ljkwisundenedg=fg. 24

PAGE 30

4.4Possibleapplicationsandfuturedirections.Intuitively,intrinsicallynon-locallanguagesgeneratedbytransducerscanbedescribedasthenon-locallanguagesthatcanbeobtainedthroughanontrivialtrans-ducer,i.esomethingdierentthan,say,atransducerwithoutputfunctionequaltotheshifttotheright.Example1:Oneoftheexamplesofintrinsicallynon-locallanguagesisthesetofalltwo-dimensionalblockscontainingatmostone'1',i.e.,blocksoftheformB=000000000000000001000000000000000000000000Itiseasytocheckthatthereisnotransducerthatwouldproducepreciselythislanguage.Howeveritispossibletoconstructasimilarlanguagebyaddingapaddingtoalatterone,forexample300000030000002001000100000010000001000000SuchlanguagewouldberecognizablebytransducerdepictedonFigure6.Foragivensetoftiles,whichcanbethoughtofasasetofrectangles,letsconsiderthesetofall2dimensionalblocksthatcouldbeconstructedfromthetileswithoutrotation,andconsidertheproblemofdeterminingifthereisatransducer,forwhichL2consideswiththatset.Unfortunately,ingeneralthisisnotpossible,forexampleconsidersetoftwotiles 25

PAGE 31

q0 q3 q1 q2 1 1 3 3 2 3 1 2 1 0 0 1 0 0 0 0 0 0 Figure6:Transducergeneratingnon-locallanguage andsupposethatthereisatransducersuchthatL2equalsthesetofallpossibletwodimensionalblocksobtainedthroughtranslationsofagivensetoftiles.Asblocks101111111111and101111101111showdenotetheseblocksasC2andC3respectively,thisisnotpossiblesinceifsuchtransducerwouldexist,thenmustacceptC3[][2]andoutputword111111,asitproducesoneofthevalidtilings.Howeverthisisimpossible,sinceC3[][2]=C2[][2],andthusoutputs111111asathirdrowinC2,whichisnotoneofthevalidtilings.4.5EntropyofL2 Denition4.5. ForLAtheentropyofLishL=limn!1sup1 n2logjBn;nj;whereBn;n=fC2LjCisannnblockg. 26

PAGE 32

ItisclearthatentropyofL2is0foranydeterministictransducer,sinceforeachnitcontainsatmostjAjnblockstoeachw2L2correspondsuniqueblockofheightn.However,ifL2isatwodimensionallanguagecorrespondingtoanondeterministictransducer,entropymaynolongerbe0.Forexample,lettransducer=A;fqg;;fqg;fqgwith=fq;a;b;qja;b2Ag,i.e.consistsofonestatewithallpossibletransitions.HenceL2=Aandforeachn,jBn;nj=jAjn2.ThushL2=hA=logjAj.Foragiventransducerandw2Ldenedegw=jwj,thenumberofdistinctwordsthatcanbeoutputtedbyoninputw,andextendthisnotationtosetsbydegS=sups2Sfdegsg.ThenthenumberofdistinctnnblockscontainedinL2withwjwj=nasarstrowisboundedabovebythefollowingexpression:nwn)]TJ/F19 7.97 Tf 6.586 0 Td[(2Yi=0degiw:Ifforallw2Lwithjwj=nhavethatdegwnkthenhL2=limn!1sup1 n2logBn;nlimn!1sup1 n2logXjwj=nn)]TJ/F19 7.97 Tf 6.586 0 Td[(2Yi=0degiwlimn!1sup1 nlogjAjnk=0:Thusifthereexistsksuchthatforanyw2L,wjwjk,i.e.jwjhasapolynomialbound,theentropyisstillequaltozero. Proposition4.2. ForanyregularlanguageLthereissuchthatL=LandhL2=hL Proof. LetMdenoteaFSA,suchthatLM=L,andletbeitstransitionre-lation.ConstructtransducerfromMbyredeningtransitionrelationas0=fq1;a;s;q2jq1;a;q22;s2Ag.ThusL=L. 27

PAGE 33

NowconsiderL2.NotethatforB2L2,thelast,kthrowmaynotbeinL.LetBk;kdenotethesetofalltwodimensionalblocksinL2ofsizekkandB0p;kdenotethesetofallpkblocksB2L2withB[p][]2LB[p][]denotespthrowofblockB.ThusitmustbetruethatjBk;kjjB0k)]TJ/F19 7.97 Tf 6.586 0 Td[(1;kjjAjkjB0k;kjjAjk.Hence,hL2=limsupk!11 k2logjBk;kj.1limsupk!11 k2logjB0k;kjjAjk4.2=limsupk!11 k2logjB0k;kj.3Now,sincejB0k;kjjBk;kj,havethathL2=limsupk!11 k2logjB0k;kjAlso,jB0k;kj=jLAkjksinceB0k;k=fB2AjB[i][]2LAk;i=1:::kg.ThushL2=limsupk!11 klogjLAkj=hL Corollary4.2. LetLbearegularlanguage.ThentheentropyofLisanupperboundfortheentropyofL2foranytransducerwithL=L. Proof. LetL2Regandlet=A;Q;;q0;FbesuchthatL=L.Thenconsider0=A;Q;0;q0;F,with0=fq;a;s;pjs2Aandq;a;t;p2;forsomet2Ag.Hence0andhL2hL20.Bytheproofoftheproposition4.2,hL20=hLandclaimfollows. 28

PAGE 34

5ConclusionInthisworkwehaveattemptedtoanalyzesequencesthatcanbeobtainedbyiteratingatransduceronwordsofitsinputlanguage.Wecalledsuchsequencestrans-ducerrecognizable.Forthesequencesthatarenottransducerrecognizablewehavedevelopedanotionofrecognitionincontextandshownthatallniteandperiodicsequencesarerecognizedincontext.Asnotedintheintroduction,transducerrecog-nizablesequencescanberelatedtotheblocksconstructedofWangtiles.EachoftheWangtilescorrespondstoaspecictransitioninthetransducer.Whenthetransducerisdeterministic,eachtileisuniquelydeterminedbyitsleftandbottomedge.ThisisabenecialpropertywhenprocessofassemblyofWangtilesinblocksissimulatedbyDNAstrands,asithelpstoreducethenumberofincorrectpartialtilings[13].ToeachnondeterministictransducerweassociatedatwodimensionallanguageL2.InthecasewhenL2isalocalpicturelanguage,weinvestigatedpropertiesofLthatthisconditioninduces.WehaveshownarelationbetweentheentropyofL2andL.ThisrelationsuggestsawiderangeofthepossiblevaluesfortheentropyofL2.Characterizationofpatternsthatcanbegeneratedbyiterationoftransducersmaybeofinterestinapplications,inparticular,inalgorithmicself-assemblyoftwo-dimensionalarrayswithDNAtiles.Also,itmaybeofinteresttoinvestigatesomedecidabilityquestionsfortheselanguagesaswell:forexample,givenatransducergeneratedlanguage,isthereanmn-blockinthelanguageforeverym;n?Therela-tionshipoftheclassoftransducergeneratedlanguageswiththeclassofunambiguousornon-deterministicpicturelanguagesaswellastransitivityandmixingpropertiesoftheselanguagesremaintobeinvestigatedaswell. 29

PAGE 35

References[1]L.M.Adleman,MolecularComputationofSolutionstoCombinatorialProblems,Science,266,1021-1024.[2]A.V.Aho,J.D.Ullman,Thetheoryofparsing,translationandcompiling.VolumeI,Prentice-Hall,1972.[3]B.Chakraborty,N.Jonoska,N.C.Seeman,ProgrammabletranducerbyDNAself-assembly,submitted.[4]P.J.Denning,Machines,Languages,andComputation.Prentice-Hall,1978[5]E.Dolzhenko,N.Jonoska,OnComplexityofTwoDimensionalLanguagesGener-atedbyTransducers,submitted.[6]B.Grunbaum,G.C.Shephard,Tilingsandpatterns.W.H.FreemanandCom-pany,1987.[7]N.Jonoska,et.al.,TransducerswithProgrammableInputbyDNASelf-assembly,LectureNotesinComputerScience,Springer2950,219-240.[8]N.JonoskaandJ.B.Pirnot,TransitivityinTwo-DimensionalLocalLanguagesDenedbyDotSystems.InternationalJournalofFoundationsofComputerScience,17,435-464.[9]V.Manca,C.Martin-Vide,Gh.Paun,NewcomputingparadigmssuggestedbyDNAcomputing:computingbycarving,BioSystems5247-54.[10]Gh.Paun,Ontheiterationofgsmmappings,Rev.Roum.Math.PuresAppl., 30

PAGE 36

23,921-937.[11]Rozenberg,SalomaaEds.HandbookofFormalLanguages:Volume3.Be-yondWords.2002.[12]H.Wang,Notesonclasstilingproblems.FundamentalMathematics,82,295-305.[13]E.Winfree,AlgorithmicSelf-AssemblyofDNA:TheoreticalMotivationsand2DAssemblyExperiments.JournalofBiomolecularStructureandDynamics,11S2,263-270.[14]E.Winfree,F.Liu,L.A.Wenzler,N.C.Seeman,Designandself-assemblyoftwo-dimensionalDNAcrystals,Nature,394,539-544. 31


printinsert_linkshareget_appmore_horiz

Download Options

close
Choose Size
Choose file type
Cite this item close

APA

Cras ut cursus ante, a fringilla nunc. Mauris lorem nunc, cursus sit amet enim ac, vehicula vestibulum mi. Mauris viverra nisl vel enim faucibus porta. Praesent sit amet ornare diam, non finibus nulla.

MLA

Cras efficitur magna et sapien varius, luctus ullamcorper dolor convallis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Fusce sit amet justo ut erat laoreet congue sed a ante.

CHICAGO

Phasellus ornare in augue eu imperdiet. Donec malesuada sapien ante, at vehicula orci tempor molestie. Proin vitae urna elit. Pellentesque vitae nisi et diam euismod malesuada aliquet non erat.

WIKIPEDIA

Nunc fringilla dolor ut dictum placerat. Proin ac neque rutrum, consectetur ligula id, laoreet ligula. Nulla lorem massa, consectetur vitae consequat in, lobortis at dolor. Nunc sed leo odio.